Problem 77 - Algebra and Trigonometry

Question

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{2}+x $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Define Even and Odd Functions

A function is even if \(f(-x) = f(x)\) for all \(x\) in the domain. It is odd if \(f(-x) = -f(x)\) for all \(x\) in the domain.

Step 2: Calculate \(f(-x)\)

Substitute \(-x\) into the function: \(f(-x) = (-x)^2 + (-x) = x^2 - x\).

Step 3: Compare \(f(-x)\) and \(f(x)\)

The original function \(f(x) = x^2 + x\). \(f(-x) = x^2 - x\). Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the function is neither even nor odd.

Key Concepts

Even Functions

Even functions have a particular symmetry around the vertical axis, also known as the y-axis. Mathematically, a function is considered even if, for every element \(x\) in its domain, the equation \(f(-x) = f(x)\) holds true. This symmetry implies that the graph of the function mirrors itself on either side of the y-axis.

Characteristics of Even Functions:

• The graph has a reflective symmetry about the y-axis.
• Common examples include polynomials like \(x^2\), \(x^4\), \(\cos(x)\).
• All the powers of \(x\) in the polynomial are even integers.

When trying to determine if a function is even, it's a good practice to substitute \(-x\) in place of \(x\) and check whether the new expression is identical to the original function. This method is precisely what we attempted earlier in the exercise. However, for the function \(f(x) = x^2 + x\), substituting gave us \(f(-x) = x^2 - x\), which is not the same as \(f(x)\). Therefore, this function is not even.

Odd Functions

Odd functions have their own unique symmetry, specifically around the origin. For a function to be considered odd, it must satisfy the condition \(f(-x) = -f(x)\) for every \(x\) in the domain. This definition essentially means that the graph of the function is rotationally symmetric around the origin.

Characteristics of Odd Functions:

• The graph exhibits rotational symmetry about the origin at \((0,0)\).
• Examples include polynomials like \(x^3\), \(x^5\), and trigonometric functions like \(\sin(x)\).
• All the powers of \(x\) in the polynomial are odd integers.

To check whether a function is odd, we replace \(x\) with \(-x\) in the function and verify if the result is the negative of the original function. When applying this to \(f(x) = x^2 + x\), we found \(f(-x) = x^2 - x\), which is neither \(-f(x)\) nor \(f(x)\). Consequently, \(f(x)\) is not odd.

Graph Symmetry

Graph symmetry is a vital concept in understanding the visual behavior of functions. Symmetry can appear in several forms in calculus, impacting how graphs are constructed and interpreted.

Types of Graph Symmetry:

• Even Function Symmetry: Such graphs mirror themselves about the y-axis. For instance, \(y = x^2\) shows a perfect reflection across the y-axis.
• Odd Function Symmetry: These graphs have rotational symmetry centered at the origin. An example is \(y = x^3\), which looks identical if rotated 180 degrees around the origin.
• Neither Even nor Odd: Some functions do not exhibit either type of symmetry. Their graphs can be asymmetrical and do not follow the properties of even or odd function symmetry.

Identifying symmetry helps to graph functions more efficiently by reducing the amount of calculation needed. It allows for better prediction of the graph shape across the entire domain. In our given example of \(f(x) = x^2 + x\), the absence of symmetry meant its graph doesn't fall into the categories of even or odd, which is important to note when sketching or analyzing such functions.